The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system:
where is the number of independent repetitions, and is the quantum Fisher information.[1][2]
Here, is the state of the system and is the Hamiltonian of the system. When considering a unitary dynamics of the type
where is the initial state of the system, is the parameter to be estimated based on measurements on
Let us consider the decomposition of the density matrix to pure components as
The Heisenberg uncertainty relation is valid for all
From these, employing the Cauchy-Schwarz inequality we arrive at [3]
Here [4]
is the error propagation formula, which roughly tells us how well can be estimated by measuring Moreover, the convex roof of the variance is given as[5][6]
where is the quantum Fisher information.